Aristotle has a separate work called On the Heavens, De Caelo, in which he doesn't so much talk about Eudoxus' model as he backs off and spends some time doing essentially philosophic reasoning about what has to be true of the celestial run. And I'm just going to run through these items because they became very important in, right down, essentially down to Copernicus and Kepler.

The first principle is the celestial realm is filled with a different kind of matter entirely from earth, fire, air, and water, the four elements. The fifth element is ether. The four elements have a natural motion toward the middle of the earth. The fifth element does not have a natural motion toward the middle of the earth, it's natural motion is uniform circular motion.

Which is eternal and self perpetual and it can't according to Aristotle speed up and slow down because something would have to make it slow down and then speed up and there's nothing to do that. So it's self sustaining perfect circular motion all the time. The Earth has to be in the middle of all of this.

Because if it were not and you had spheres like Eudoxus' spheres they would tend to fall toward the Earth. If they're centered perfectly on the center, then any tendency they could conceivably have toward falling toward the middle Is offset, okay? The other reason the earth has to be in the center is the so-called sublunary realm, that is the realm below the celestial realm.

Everything falls toward the center of the earth. And that explains why the earth has to be a sphere. Get rid of one of these myths. No one seriously from Aristotle forward thought the Earth was flat, okay? That's just one of these nice myths that we get when we learn about Columbus.

But it was quite well-recognized by everybody in astronomy. And doesn't take much to realize this like it's a shadow of the moon. A shadow of the moon as it goes across the sun on the solar eclipse. It's not on a straight line. I did that wrong. Shadow of the earth going across the sun and the shape on the moon.

So both the moon and we know the moon is a sphere, but of course the Earth has to be spherical because it's circular. And the other thing, the Earth is a very small sphere compared to the celestial realm. But Aristotle has all the stars sitting on a single sphere, rotating daily.

And that sphere's motion, in a kind of gear fashion, produces the motion of all the others and maintains it. Okay, that was the comment before about that being the prime mover, the motion of the stars. And the unmoved mover started it all. That's why it's in Book Lambda.

He calls attention, and I just wanna note he gives answers to both of these. But he says there are puzzles about all of this. And the two he singles out are worth actually noting just in passing. One of the strangest is the problem why we find the greatest number of movements in the intermediate bodies.

And not, rather, in each successive body, a variety of movement proportionate to its distance from the primary motion. That is, you'd think the amount of motion would increase monotonically toward the Earth or away from the Earth. It doesn't. The intermediate have the most motion. The moon has less and the stars have less altogether, why?

Then correspondingly a puzzle, why should there be so many stars out there? The whole array seems to defy counting, while of the other stars meaning the planets and the sun and the moon. Each one is separated off. In no case do we find two or more attached to the same motion.

So why do we have five of these objects moving completely individually, and literally thousands of them moving in unison with one another? And he comes up with classic philosophic arguments for why we might accept that. The important thing and the reason I put this slide up here is the two-fold motion.

The earth is motionless in the center, two-fold principles, and second, all celestial motion is uniform circular motion, because it is self-perpetuating, and nothing else is. Fair enough? That's a background to everything else. Now I'm just setting this up, the division here, Eudoxus and Aristotle. Eudoxus overlapped Plato and Aristotle, and was in Athens when they were both alive.

So Eudoxus is a common figure to the three of them. The gap I put in now is to separate what's called the Hellenic period from the Hellenistic. That's normally thought of as before and after Alexander. Before Alexander, Greece had its city-states. They fought with one another, but they didn't dominate all of Asia Minor and all of the Mediterranean.

After Alexander conquered everything, it was different. So you'll see the four great mathematicians of the Hellenistic period, Euclid, Archimedes, Apollonius, and Hipparchus. None of them lived in Greece proper. But they are very much Greek writing, Greek speaking, et cetera. Couple comments about each. Euclid, of course, you all know what he did in producing the elements was not to generate all of that.

He's instead collecting work done over two centuries before him and giving it an organization in an axiomatic form. And because he did it so well, nobody bothered to preserve the stuff before him. That's why Eudoxus' work is lost, but what we know of it comes from Euclid. Euclid also did some work on motion, a fair amount of work on optics, and some work in astronomy.

And he thrived in Alexandria during the period just after Alexander. Archimedes is from Syracuse which, of course, is over in Sicily. Lived after, barely overlapping Euclid. He's a major mathematician. He's gonna be important in this course cuz Galileo modeled all of his science off of Archimedes. He's of course famous for the amount of water displaced by anybody.

Namely it's weight. And that's one of the central works that Galileo deals with. We'll come back to him later. Apollonius is legendary. We have a most important work of his, the Conics. He worked out in Euclidian form the complete geometry of parabolas, hyperbolas, and ellipses. Not with equations as we do today.

Altogether geometrically. They are the sections if you take a plane and cut it through a cone, you get either a hyperbola, an ellipse or a parabola. And you now develop the properties off of the cone itself in that angle. The complete theory of conics is there. Kepler read the conics at the time he was first beginning work, which we'll see in two weeks on planetary motion.

That's how he was so familiar with ellipses etc. That's the importance of Apollonius historically. He also made a very major contribution to Astronomy. He became the first real proponent of going to epicycles. To capture the retrograde motion rather than nested spheres. We'll get to that in just a moment.

That work has not survived. We know of it only from Ptolemy. What Apollonius did, did not really work it didn't capture the variable motion of the retrograde loops at all. You'll see why in just a moment at a subsequent slide. Coming along after him, Hipparchus had a major advantage over him.

Hipparchus had real access to apparently virtually all the Babylonian data. And he realized right away that what Apollonius' system of epicycles didn't work at all. So, he tried to develop a system that would work. It's a complex combination. It has epicycles and it has observers off-center. We'll get to it in just a moment.

But he then discovered it didn't work either, and he didn't have the math to continue. We have none of Hipparchus' work because everything we know is in Ptolemy's Almagest because that's where Ptolemy starts. He starts form Hipparchus and builds off of that, constantly crediting Hipparchus. But notice a 300 year gap between Hipparchus and Apollonius.

And you can naturally ask, well all this stuff takes place between 323 BC and 120 BC, all this activity, and then we get a 300 year hiatus before we get to Ptolemy. Why do we get a 300 year hiatus? Answer, during those 300 years, spherical trigonometry was developed.

That's what Hipparchus did not have the mathematics to do anything beyond simple work on a sphere. Again, we know one name Menelaus, because it's Menelaus's theorem for spherical trigonometry. But the development of spherical trigonometry we know almost entirely from what Ptolemy tells us. Ptolemy did to all of this work what Euclid did to all prior geometry, namely cause nobody to keep it.